Kac's Walk on Rotation Matrices Mixes in Θ(n2) Steps: A Proof Discovered with AI
Abstract
Let N=binom n2=dimSO(n). We prove that the coordinate-plane Kac walk on SO(n) has total-variation mixing time of order N: for every fixed 0<ε<1, tmix(n)(ε)=Θε(n2). The lower bound is the dimensional singularity obstruction before N steps. The upper bound removes the final logarithm from the previously known O(n2log n) estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full N−1 resource; mixed packets contain a typed O(n−1/2) Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an O(n) squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is O(n/c2) after cN steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.
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BibTeX
@misc{liu2026kacs,
title = {{Kac's Walk on Rotation Matrices Mixes in $\boldsymbolΘ(n^2)$ Steps: A Proof Discovered with AI}},
author = {Tianle Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29403},
eprint = {2608.29403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29403},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kac's Walk on Rotation Matrices Mixes in Θ(n^2) Steps: A Proof Discovered with AI
AU - Tianle Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.29403
UR - https://arxiv.org/abs/2608.29403
AB - Let N=binom n2=dimSO(n). We prove that the coordinate-plane Kac walk on SO(n) has total-variation mixing time of order N: for every fixed 0<ε<1, t_mix^(n)(ε)=Θ_ε(n^2). The lower bound is the dimensional singularity obstruction before N steps. The upper bound removes the final logarithm from the previously known O(n^2log n) estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full N^−1 resource; mixed packets contain a typed O(n^−1/2) Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an O(n) squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is O(n/c^2) after cN steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.
ER -